Charmageddon - I work for the police, have done for 10 years. A few facts (with the benefit of knowing police regs and the law)
Custody roles - custody officers (the people who do the checks/meals etc etc) are police staff, not warranted officers. The Custody desk and booking in is a Sgt as there's something called a 'necessity test' which has to be conducted by a warranted officer with those powers of detention.
Schools - is done by PCSOs or Specials who are police staff. You might have heard of radicalisation/prevent/ASB/cyber bullying/grooming to name a few. So yes, we need to be in schools.
cycling - used to be done by OFF DUTY cops out of personal interest
Mental health/child welfare -apart from the fact that the NHS/mental health teams and social services have also been trashed, there's actually quite a lot of legislation around all of these which believe it or not requires police powers - Google Section 136, for instance, or the difficult Section 135. So whilst at an uneducated look, it's not the 'police' job (think missing persons - why do the police look and not just the family? Really think about it) actually they usually have a massive legal aspect and require those pesky warranted powers.
Desk jobs/admin - little secret - police officers can't be made redundant. They are not 'employed' but serve the crown by appointment. Therefore - to save money and 'protect the front line' what has been happening is as follows:
The Tories got in and slashed budgets. Every force immediately froze recruitment. It takes 2 years to train a new PC. Meanwhile officers either retired, or as we're seeing more and more, left for other careers (they trashed the pension and so for some added 20 years to their service for less money so it wasn't worth staying bearing in mind the pay freeze and increasing workloads).
Unfortunately, stopping recruitment didn't save enough money, nor did not replacing those leaving. So then they started trying to force retirements. Then they realised the way to save was to make police staff - the people you suggest do the things you've mentioned (there are TONS of others you probably haven't thought about - basically anything which doesn't require arrest powers) redundant. So this they did with gusto.
The police can't 'go into the red' so these redundancies were huge and brutal. Hundreds lost their jobs (bear with me). Those jobs however, didn't go away. Ie - the investigation caseload carried by police staff investigators didn't go away. The training and ongoing LEGAL maintenance of things like use of force, first aid etc didn't go away. That got worse, because so few officers were on duty at any given time, it was difficult to get away for the training, so double or triple the amount of courses needed to be run to ensure compliance. CPS didn't reduce their need for court files to be completed and disseminated, officers called to court, and victims still needed supporting. Officers still needed assigning to shifts, covering operations against serious crime, carrying out Intel duties. They still needed paying, their tax done, their stolen days off rescheduled (to be cancelled again). Judges still insisted on the officer who investigated the case going to court and running backwards and forwards for more evidence.
But the police staff had gone. So they had to move warranted police officers into those roles or add them to their 'day job' because the person had gone but not the demand for what they did.
So, kind of, in a bullshitty, smoke and mirrors way, they'd 'protected the front line' - no cops had been made redundant, because legally they can't be made redundant.
Your idea is fab 'we don't need more police, we need more civvies' but you see, the government forced those very people out!
It's a bit like everyone now saying 'why did the police let the terrorists go' without any knowledge of law, proportionality and human rights legislation (Google RIPA) you simply cannot level facile judgements of what should be done.
The Tories hate public services. Luckily, us in public services don't hate the public, and against the background off 'more for less' on all fronts, we carry on doing our jobs. We do it because in various ways we want to help our fellow human beings.
Life isn't just about the worry of getting taxed a bit more. I hope you never get to realise that as we see it every day. Attend a few suicides, then think about your vote.